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%\title{Python 数学实验与建模} 
\title{伊藤引理}
%(1.1-1.2) 
%\institute{上海立信会计金融学院}
\author{{\ppr LQW}}
\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
%\date{{\ppr 2023年1月6日} }

\maketitle

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\begin{frame}{内容提要 }

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\begin{enumerate}
\item[6.1.]  求导的链式法则
\item[6.2.]  泰勒公式
\item[6.3.]  伊藤引理的简单版本
\item[6.8.]  伊藤指数、几何布朗运动
%\item  伊藤过程
%\item  伊藤引理的复杂版本

\end{enumerate}

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\begin{frame}{6.1. 求导的链式法则 }

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\begin{itemize}\itemsep0.5em

\item  设 $y=f(x)$ 和 $x=g(t)$ 都是可微函数。则复合函数的求导法则为 $$[f(g(t))]' = f\,'(g(t))g\,'(t).$$

\item  链式法则的积分形式为 $$f(g(t)) - f(g(0)) = \int_0^t f\,'(g(s)) g\,'(s)ds = \int_0^t f\, (g(s)) dg(s). $$

\end{itemize}

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\begin{frame}{6.2. 泰勒公式 }

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\begin{itemize}\itemsep0.5em

\item  设 $y=f(x)$ 和 $x=g(t)$ 都是可微函数。
\item  问题：复合函数 $f(g(t))$ 的微分是什么？
\item  回答：根据泰勒公式，有
\begin{eqnarray*}
df(g(t)) &=& f(g(t+dt)) - f(g(t)) \\
&=& f\,'(g(t)) d(g(t)) + \frac{1}{2!} f\, ''(g(t)) (dg(t))^2 + \frac{1}{3!} f\, '''(g(t)) (dg(t))^3 + \cdots 
\end{eqnarray*}

\end{itemize}

\end{frame}

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\begin{frame}{6.3. 伊藤引理的第一种形式 }

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\begin{itemize}\itemsep0.5em

\item  设 $\{B_t,\, t\ge 0\}$ 是布朗运动。记增量 $dB_t = B_{t+dt} - B_t$. 
\item  设 $f(x)$ 是一个二次可微函数。 
\item  问题：复合函数 $f(B_t)$ 的微分是什么？
\item  回答：根据泰勒公式，有 
\begin{eqnarray*}
df(B_t) 
&=& f(B_{t+dt}) - f(B_t) \\ 
&=& f(B_t + dB_t) - f(B_t) \\ 
&=& f\,'(B_t)dB_t + \frac{1}{2}f\,''(B_t)(dB_t)^2 + \cdots  \\ 
&=& f\,'(B_t)dB_t + \frac{1}{2} f\,''(B_t)dt + \cdots
\end{eqnarray*}

\end{itemize}

\end{frame}

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\begin{frame}{6.4. 伊藤引理的积分形式 }

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\begin{itemize}\itemsep0.5em

\item  {\color{red}在等式两边同时对 $t$ 积分，可得
\begin{eqnarray*}
f(B_t) - f(B_s) = \int_s^t df(B_x) = \int_s^t f\, (B_x)dB_x + \frac{1}{2} \int_s^t f\, ''(B_x)dx. 
\end{eqnarray*}
}

\item  注：我们认为 $(dB_t)^3 = (dB_t)dt$ 是比 $dt$ 更高阶的无穷小量，所以看成零。


\end{itemize}

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\begin{frame}{6.5. 伊藤引理的应用 - 1 }

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\begin{itemize}\itemsep0.5em

\item  设 $f(x)=x^2$. 代入伊藤引理的等式，可得
\begin{eqnarray*}
B_t^2 - B_s^2 = f(B_t) - f(B_s)  &=& \int_s^t f\,' (B_x)dB_x + \frac{1}{2} \int_s^t f\, ''(B_x)dx \\
&=& \int_s^t 2B_xdB_x + \frac{1}{2} \int_s^t 2dx \\
&=&2\int_s^t B_xdB_x + t-s. 
\end{eqnarray*}

\item  因此求得一个伊藤积分的具体结果：
\begin{eqnarray*}
\int_s^t B_xdB_x = \frac{1}{2} \left[ (B_t^2 - B_s^2) - (t-s) \right]. 
\end{eqnarray*}

\end{itemize}

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\begin{frame}{6.6. 伊藤引理的应用 - 2 }

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\begin{itemize}\itemsep0.5em

\item  设 $f(x)=x^3$. 则 $f\,'(x)=3x^2$, $f\,''(x)=6x$. 代入伊藤引理的等式，可得
\begin{eqnarray*}
B_t^3 - B_s^3 = f(B_t) - f(B_s)  &=& \int_s^t f\,' (B_x)dB_x + \frac{1}{2} \int_s^t f\, ''(B_x)dx \\
&=& \int_s^t 3B_x^2dB_x + \frac{1}{2} \int_s^t 6B_xdx \\
&=& 3\int_s^t B_x^2dB_x + 3 \int_s^t B_xdx. 
\end{eqnarray*}

\item  但是如何用布朗运动表示上式右边的第二个积分？ $$\int_s^t B_xdx =  $$

\end{itemize}

\end{frame}

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\begin{frame}{6.7. 伊藤引理的应用 - 3 }

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\begin{itemize}\itemsep1em

\item  设 $f(x)=\exp(x)$. 则 $f\,'(x)=\exp(x)$, $f\,''(x)=\exp(x)$. 由伊藤引理可得
\begin{eqnarray*}
\exp(B_t) - \exp(B_s) = f(B_t) - f(B_s)  &=& \int_s^t f\,' (B_x)dB_x + \frac{1}{2} \int_s^t f\, ''(B_x)dx \\
&=& \int_s^t \exp(B_x)dB_x + \frac{1}{2} \int_s^t \exp(B_x)dx. 
\end{eqnarray*}

\end{itemize}

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\begin{frame}{6.8. 伊藤指数 }

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\begin{itemize}\itemsep0.5em

\item  指数函数 $f(x)= ce^x$ 是下述积分方程的解函数 $$f(t) - f(s) = \int_s^t f(x)dx.$$

\item  {\color{red}我们称符合下述等式的随机过程 $\{X_t,t\ge 0\}$ 为伊藤指数：$$X_t - X_s = \int_t^s X_xdB_x.$$}

\item  随机过程 $\{\exp(B_x), x\ge 0\}$ 不是伊藤指数。

\end{itemize}

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\begin{itemize}\itemsep0.5em

\item  设二元函数 $f(t,x)$ 有连续的二阶偏导数，一阶偏导数记为 $f_1$ 和 $f_2$, 二阶偏导数记为 $f_{11}, f_{12}$ 和 $f_{22}$. 则有泰勒公式
\begin{eqnarray*}
f(t+dt,x+dx) - f(t,x)  &=& f_1(t,x)dt + f_2(t,x)dx + \frac{1}{2} f_{11}(t,x)(dt)^2 \\
&& + f_{12}(t,x)(dt)(dx) + \frac{1}{2}f_{22}(t,x)(dx)^2 +\cdots.
\end{eqnarray*}

\item  用 $B_t$ 代替 $x$, 并注意到 $(dB_t)^2=dt$, 可得
\begin{eqnarray*}
f(t,B_t) - f(s,B_s) = f_1(t,B_t)dt + f_2(t,B_t)dB_t + \frac{1}{2}f_{22}(t,B_t)dt.
\end{eqnarray*}

\end{itemize}

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\begin{itemize}\itemsep0.5em

\item  随机过程 $\{X_t=\exp[(c-\frac{1}{2}\sigma^2)t+\sigma B_t],\, t\ge 0\}$ 符合随机微分方程
$$X_t - X_0 = c\int_0^t X_sds + \sigma \int_0^t X_sdB_s.$$

\item  证明：取 $f(t,x) = \exp[(c-\frac{1}{2}\sigma^2)t + \sigma x]$. 则 $X_t = f(t,B_t)$. 由伊藤引理即得这个称为随机微分方程的等式。

\item  注：随机过程 $\{X_t\}$ 称为几何布朗运动。

\item  注：上述随机微分方程可写成这个形式，使其看起来更像“微分方程”，
$$dX_t = cX_tdt + \sigma X_tdB_t.$$

\end{itemize}

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\begin{itemize}\itemsep1em

\item  若随机过程 $\{X_t,\, t\ge 0\}$ 由下述表达式给出，则称为伊藤过程， 
\begin{eqnarray*}
X_t &=& X_0 + \int_0^t A_s^{(1)}ds + \int_0^t A_s^{(2)}dB_s, \\
dX_t &=& A_t^{(1)}dt + A_t^{(2)}dB_t. 
\end{eqnarray*}

\item  设 $f(t,x)$ 有连续的二阶偏导数。把下述公式写完：
\begin{eqnarray*}
f(t,X_t) - f(s,X_s) &=& f_1(t,X_t)dt + f_2(t,X_t)dX_t + \frac{1}{2}f_{11}(t,X_t)(dt)^2 \\ 
&&+ f_{12}(t,X_t)(dt)(dX_t) +\frac{1}{2}f_{22}(t,X_t)(dX_t)^2 + \cdots.
\end{eqnarray*}

\end{itemize}

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\begin{thebibliography}{99}

\bibitem{mikosch} {\ppr Thomas Mikosch}. {\ppr Elementary Stochastic Calculus}. 世界图书出版公司，{\ppr 2009} 年 {\ppr 8} 月第 {\ppr 1} 版。
\bibitem{wangjun} 王军、邵吉光、王娟. 随机过程及其在金融领域中的应用. 清华大学出版社，北京交通大学出版社，{\ppr 2018} 年{\ppr 8} 月第 {\ppr 2} 版。
\bibitem{zhangbo} 张波、商豪. 应用随机过程. 中国人民大学出版社，{\ppr 2016} 年 {\ppr 6} 月第 {\ppr 4} 版。
%\bibitem{karlin} {\ppr Mark A. Pinsky, Samuel Karlin}. {\ppr An Introduction to Stochastic Modeling}. 机械工业出版社，{\ppr 2013} 年 {\ppr 2} 月第 {\ppr 1} 版。


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